## Calculation of the power factor

### Calculation of the power factor

#### Chapter 11 - Power Factor

As mentioned earlier, the angle of this "power triangle" graphically shows the relationship between the amount of power consumed (or consumed) and the amount of power absorbed / returned. It also happens to be the same angle as the impedance of the circuit in polar form. When expressed as a fraction, this ratio between real power and apparent power is called the power factor for this circuit. Since the real power and the apparent power form the neighboring or hypotenus sides of a right triangle, the power factor ratio is also equal to the cosine of this phase angle. Use values ​​from the last example circuit:

It should be noted that the power factor, like all ratio measurements, is a unitless quantity.

For the purely resistive circuit, the power factor is 1 (perfect) because the reactive power is zero. Here the energy triangle would look like a horizontal line because the opposite (reactive power) side would have a length of zero.

For the purely inductive circuit, the power factor is zero because the true power is zero. Here the energy triangle would look like a vertical line because the adjacent (true energy) side would have a length of zero.

The same could be said for a purely capacitive circuit. If there are no dissipative (ohmic) components in the circuit, then the true power must be zero, which makes any power in the circuit purely reactive. The power triangle for a purely capacitive circuit would again be a vertical line (downwards instead of upwards as in the purely inductive circuit).

Power factor can be an important aspect to consider in an AC circuit because a power factor less than 1 means the circuit wiring must carry more current than would be required if the circuit had zero reactance) current to the resistive load. If our last example circuit had been purely resistive, we would have been able to deliver a full 169.256 watts to the load at the same 1. 410 amperes, instead of the 119.365 watts it currently dissipates at the same amperage . The poor power factor makes the power system inefficient.

A poor power factor can paradoxically be corrected by adding another load to the circuit that draws an equal and opposite amount of reactive power to offset the effects of the inductive reactance of the load. The inductive reactance can only be canceled by the capacitive reactance, so we need to add a capacitor in parallel to our example circuit as an additional load. The effect of these two opposite reactances in parallel is to make the total impedance of the circuit equal to its total resistance (to make the impedance phase angle equal to or at least closer to zero).

Since we know that the (uncorrected) reactive power is 119,998 VAR (inductive), we need to calculate the correct capacitor size to produce the same amount of (capacitive) reactive power. Since this capacitor is directly in parallel with the source (known voltage), we will use the power formula based on voltage and reactance:

Let's use a rounded capacitor value of 22 μF and see what happens to our circuit: (picture below)

The parallel capacitor corrects the lagging power factor of the inductive load. V2 and node numbers: 0, 1, 2 and 3 are SPICE-related and can be ignored for the moment.

The overall power factor for the circuit has been significantly improved. The main current has been reduced from 1.41 amps to 994.7 milliamps, while the power dissipation at the load resistor remains unchanged at 119.365 watts. The power factor is much closer to 1:

Since the impedance angle is still a positive number, we know that the circuit as a whole is still more inductive than capacitive. If our power factor correction efforts had been perfectly on target, we would have become ohmic at an impedance angle of exactly zero or purely ohmic. If we had added too large a capacitor in parallel we would have got an impedance angle that was negative, indicating that the circuit was more capacitive than inductive.

A SPICE simulation of the circuit of (figure above) shows that the total voltage and the total current are almost in phase. The SPICE circuit file has a zero volt voltage source (V2) in series with the capacitor so that the capacitor current can be measured. The start time of 200 ms (instead of 0) in the transient analysis instruction allows the DC conditions to stabilize before data is collected. See SPICE list "pf.cir power factor".

pf.cir power factor V1 1 0 sin (0 170 60) C1 1 3 22uF v2 3 0 0 L1 1 2 160mH R1 2 0 60 # resolution stop start .tran 1m 200m 160m .end

The muscle diagram of the various currents in relation to the applied voltage V. total is shown in (figure below). The reference is V totalwith which all other measurements are compared. This is because the applied voltage V total occurs on the parallel branches of the circuit. There is no common power for all components. We can compare these currents with V totally compare .

Zero phase angle due to in-phase V total and I. total . Chasing I L. in relation to V total is indicated by a leading I. C. corrected.

Note that the total current (I. total ) in phase with the applied voltage (V total ) is what indicates a phase angle close to zero. It's not a coincidence. Note that the lagging current IL of the inductor would have caused the total current to have a lagging phase somewhere between (I. total ) and I. L has . The leading capacitor current I C. however, it compensates for the delayed inductor current. The result is a total current phase angle somewhere between the inductor and capacitor currents. In addition, this total current (I. total ) by calculating a suitable capacitor value forcibly in phase with the entire applied voltage (V total ) brought.

Since the total voltage and current are in phase, the product of these two waveforms, power, will always be positive during a 60 Hz cycle, as shown in Figure above. Had the phase angle not been corrected to zero (PF = 1) the product would have been negative, where positive parts of one waveform overlapped negative parts of the other as in the figure above. Negative energy is fed back to the generator. It can't be sold; However, it consumes electricity in the resistance of the electrical lines between the load and the generator. The parallel capacitor corrects this problem.

Note that the line loss reduction applies to lines from the generator up to the point where the capacitor is applied for power factor correction. In other words, there is still current circulating between the capacitor and the inductive load. This is usually not a problem as power factor correction is applied close to the offending load, similar to an induction motor.

It should be noted that too much capacitance in an AC circuit results in a low power factor as well as too much inductance. You need to be careful not to overcorrect when adding capacitance to an AC circuit. You also need to be very careful to use the correct capacitors for the job (adequate for line voltages and the occasional spike from lightning strikes, for continuous AC service, and able to handle the expected amperages).

If a circuit is predominantly inductive, we say that its power factor is lagging (because the current wave for the circuit lags behind the applied voltage wave). Conversely, if a circuit is predominantly capacitive, we say that its power factor is leading. So our example circuit started with a power factor of 0.705 and was corrected to a power factor of 0.999.

• REVIEW:
• Poor power factor in an AC circuit can be "corrected" or restored to a value close to 1 by adding a parallel reactance that opposes the effect of the reactance of the load. If the reactance of the load is inductive (which is almost always the case), a parallel capacitance is required to correct for a poor power factor.